Date & Time: Monday, June 27, 2011, 16:00 - 17:00

Venue:Ramanujan Hall, Department of Mathematics

Title: Metric Projections in Spaces of Continuous Functions

Speaker: Professor A.L. Brown of the University College London (U.K.)

Abstract: Let T be a topological space (a compact subspace of Rm , say) and let C(T ) be the space of real continuous functions on T , equipped with the uniform norm: f = maxt∈T |f (t)| for all f ∈ C(T ). Let G be a finite dimensional linear subspace of C(T ). If f ∈ C(T ) then d(f, G) = inf{ f − g : g ∈ G} is the distance of f from G, and PG (f ) = {g ∈ G : f − g = d(f, G)} is the set of best approximations to f from G. Then PG : C(T ) → P(G) is the set-valued metric projection of C(T ) onto G. In the 1850s P. L. Chebyshev considered T = [a, b] and G the space of polynomials of degree ≤ n − 1. Our concern is with possible properties of PG . The historical development, beginning with Chebyshev, Haar (1918) and Mairhuber (1956), and the present state of knowledge will be outlined. New results will demonstrate that the story is still incomplete.